When the men of the town rose early in the morning, behold, the altar of Baal was broken down, and the Asherah beside it was cut down, and the second bull was offered on the altar that had been built. And they said to one another, "Who has done this thing?" And after they had searched and inquired, they said, "Gideon the son of Joash has done this thing." Then the men of the town said to Joash, "Bring out your son, that he may die, for he has broken down the altar of Baal and cut down the Asherah beside it." But Joash said to all who stood against him, "Will you contend for Baal? Or will you save him? Whoever contends for him shall be put to death by morning. If he is a god, let him contend for himself, because his altar has been broken down." Therefore on that day Gideon was called Jerubbaal, that is to say, "Let Baal contend against him," because he broke down his altar. {Judges 6:28-32 (ESV)}

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I want to add that the name Jerubbaal seemed fitting for that event and the imminent battles. Had the Lord given him over to the enemy, then his people would have assigned the victory to Baal and believed Baal had revenged the desecration of his alter. Now that the Lord gave victory to Jerubbal, Baal was not able to contend with the Lord's servant. How could that town not witness the Lord is God. Always look for how Yahweh himself uses us in situations to bring glory to his name.
In those days Israel had no king; and everyone did as they saw fit.

Therefore, it has a strong connotation with violence and destruction. It's the verb used in Isaiah 10:33 to describe the destruction God will inflict on Israel's enemies:

Lo! The Sovereign Lord of Hosts
Will hew off the tree-crowns with an ax:
The tall ones shall be felled,
The lofty ones cut down:

The thickets of the forest shall be hacked away with iron,
And the Lebanon trees shall fall in their majesty.—Isaiah 10:33–34 (NJPS)

It also conforms with the identity provided by the angel:

The angel of the Lord appeared to him and said to him, “The Lord is with you, valiant warrior!”—Judges 6:12 (NJPS)

Of course, when you read the story, Gideon doesn't conform at all to this identity until he overhears the dream about the barley cake. He's fearful even in his great act of defiance:

So Gideon took ten of his servants and did as the Lord had told him; but as he was afraid to do it by day, on account of his father’s household and the townspeople, he did it by night.—Judges 6:27 (NJPS)

Even so, he obeyed God and received his new name. Over the course of his lifetime, it seems that Gideon was known by the name associated with violence and that he grew into it. We read, for instance, that he aggressively pursued Midianite kings and, when his son was reluctant to do so, executed them. He also punished two towns that failed to provide aid to his army and was asked to be a king.

According to the text, Gideon refused to become a king, but in Judges 9, one of his 70 sons did establish himself as a king over at least over Shechem. During that entire story, Jerubbaal is the name used for Abimelech's father. His legacy, therefore, was not as a conqueror, but as one who struggles against foreign gods.

As it happens, Gideon himself set up an idol:

Gideon made an ephod of this gold and set it up in his own town of Ophrah. There all Israel went astray after it, and it became a snare to Gideon and his household.—Judges 8:27 (NJPS)

Conclusion

The name Gideon seems to be associated with the man's struggles and victories over foreign nations and the name Jerubbaal is associated with his struggles against idolatry. Also his military successes turned sour by the next generation, his legacy continues as a man who was faithful to God.